A variaonal frameworkforﬂow op’mizaon :anintroduc’ontoadjointssites.poli.usp.br/org/eptt2012/Presentations/Schmid/EPTT_Lecture2.pdf · time-periodic and generally time-dependent

A varia'onal framework for flow op'miza'on: an introduc'on to adjoints

Peter Schmid Laboratoire d’Hydrodynamique (LadHyX)

CNRS-‐Ecole Polytechnique, Palaiseau

EPTT, Sao Paulo, Sept. 2012

recap: time-periodic flow

Generalizations

d

dtq = L(t)q L(t + T ) = L(t) period T

with the formal solution q(t) = A(t)q0 initial condition propagator

A(t + T ) = A(t) A(T ) = A(t) C

monodromy matrix (mapping over one period)

from periodicity

qn = C qn−1 = Cn q0initial state


Generalizations

Example: pulsatile channel flow

transient growth

asymptotic contractivity



Generalizations


transient growth

asymptotic contractivity

start

large inter-period amplification



time-periodic and generally time-dependent flow

Generalizations

pseudo-Floquet analysis adjoint analysis


Can we analyze the amplification of energy between one period, i.e., for a non-periodic system matrix ?

d

dtq = L(t)q

q(t) = A(t) q0

We have

with the formal solution initial condition propagator final solution



Generalizations



We can formulate the optimal amplification of energy as

G(t)2 = maxq0

�q, q��q0, q0�

= maxq0

�A(t)q0, A(t)q0��q0, q0�

= maxq0

�AH(t)A(t)q0, q0��q0, q0�



Generalizations



G(t)2 = maxq0

�AH(t)A(t)q0, q0��q0, q0�

is a normal matrix AHA

the maximum is achieved for the principal eigenvector of AHA

the principal eigenvector (and eigenvalue) can be found by power iteration

q(n+1)0 = ρ(n)AHA q(n)

0



Generalizations




0

break the power iteration into two pieces

w(t) = A q(n)0

propagation of initial condition forward in time

first step



Generalizations




0

break the power iteration into two pieces

propagation of final condition backward in time

second step q(n+1)0 = ρ(n)AH(t)w(t)



Generalizations




0

q(n)0 A

w(t) = Aq(n)0

AHAHAq(n)0

ρ(n) direct problem

adjoint problem

scaling

updating



Generalizations



q(n)0 A

w(t) = Aq(n)0

AHAHAq(n)0

ρ(n) direct problem

adjoint problem

scaling

updating

can be any discretized solution operator. The above technique (adjoint looping) can be applied to general time-dependent stability problems. A



Generalizations



applying adjoint looping to the pulsatile (inter-period) stability problem

one period

significant inter-period transient energy growth



Generalizations


Another look at the direct-adjoint system

q(n)0 A

w(t) = Aq(n)0

AHAHAq(n)0

direct problem

adjoint problem

flow information

sensitivity/gradient information

EPTT, Sao Paulo, Sept. 2012 I −AHA correction


Generalizations


reformulate the optimal growth problem variationally

we wish to optimize J =

�q�2

�q0�2→ max

subject to the constraint d

dtq − Lq = 0



Generalizations


rather than substituting the constraint directly into the cost functional …

d

dtq − Lq = 0

J =�q�2

�q0�2=� exp(tL)q0�2

�q0�2→ max


only valid for LTI systems


Generalizations


… we enforce the equation via a Lagrange multiplier

J =�q�2

�q0�2−

�q,

�d

dtq − Lq

��→ max

q

This has the advantage that the solution to the governing equation does not have to be known explicitly.

Other constraints (such as initial and boundary conditions) can be added.



Generalizations


for an optimum we have to require all first variations of J to be zero

δJ

δq= 0

δJ

δq= 0

J =�q�2

�q0�2−

�q,

�d

dtq − Lq

��→ max

�δq,

�d

dtq − Lq

��= 0

�q,

�d

dtδq − L δq

��= 0



Generalizations



δJ

δq= 0

δJ

δq= 0

J =�q�2

�q0�2−

�q,

�d

dtq − Lq

��→ max

�q,

�d

dtδq − L δq

��= 0

d

dtq − Lq = 0



Generalizations



δJ

δq= 0

δJ

δq= 0

J =�q�2

�q0�2−

�q,

�d

dtq − Lq

��→ max

d

dtq − Lq = 0

��− d

dtq − LH q

�, δq

�= 0



Generalizations



δJ

δq= 0

δJ

δq= 0

J =�q�2

�q0�2−

�q,

�d

dtq − Lq

��→ max

d

dtq − Lq = 0

− d

dtq − LH q = 0

direct problem

adjoint problem

EPTT, Sao Paulo, Sept. 2012 KKT-condition


Generalizations


adjoint variables can be interpreted as sensitivities

J = obj−�

q,

�d

dtq − Lq

��→ max

d

dtq − Lq = f

external force

δJ = −�q, δf�

let us add an external body force to the governing equations



Generalizations


adjoint variables can be interpreted as sensitivities

J = obj−�

q,

�d

dtq − Lq

��→ max

d

dtq − Lq = f

external force

let us add an external body force to the governing equations

∇fJ = −qsensitivity to external body force



Generalizations


Example: which adjoint variable measures the sensitivity to a mass source/sink?

enforcing momentum conservation

enforcing mass conservation


−�ξ,∇ · u� �∇ξ, δu�integration

by parts

ξ is the adjoint pressure

J = obj− �u, NS(u)� − �ξ,∇ · u�


Generalizations


Example: which adjoint variable measures the sensitivity to a mass source/sink?

enforcing momentum conservation

enforcing mass conservation


∇ · u = Q δJ = �ξ, δQ�assuming a mass source/sink

adjoint pressure = sensitivity to a mass source/sink

J = obj− �u, NS(u)� − �ξ,∇ · u�


Generalizations


for the incompressible Navier-Stokes equations

∇ · u = Q

u = uw on y = 0

∇FJ = u

∇QJ = p

∂u∂t

+ advdiff(U,u) +∇p = F

∇uwJ = σ|w

forcing sensitivity


Sensitivity to internal changes (changes of specific eigenvalues with respect to parameter variations)

general formulation

A(p)q = λBq p Reynolds number wave number base-flow

Reα, β

U(y)

(A + δA)(q + δq) = (λ + δλ)B(q + δq)perturbation expansion

Generalizations



general formulation

(A + δA)(q + δq) = (λ + δλ)B(q + δq)

Generalizations


(A− λB)q + (A− λB)δq + (δA− δλB)q + (δA− δλB)δq = 00 ≈ 0

(higher order)


general formulation

(A + δA)(q + δq) = (λ + δλ)B(q + δq)

Generalizations


(A− λB)δq + (δA− δλB)q ≈ 0


general formulation

Generalizations


q+(A− λB) = 0 (A+ − λ∗B+)q+ = 0

(A− λB)δq + (δA− δλB)q ≈ 0

q+(A− λB)δq + q+(δA− δλB)q ≈ 00

use adjoint


general formulation

perturbation expansion

δλ =q+δAq

q+Bq

gradient

∇pλ =q+∇pAq

q+Bq

Generalizations


A(p)q = λBq

Example: sensitivity to a scalar parameter

Generalizations

ν = U + 2icu

∇UA = −∂x

ut = (−ν∂x + γ∂xx + µ(x))� �� A

u

λ = σ + iω

complex Ginzburg-Landau

eigenvalue sensitivity ∇Uλ = u+∇UAu

Au = λu

A+u+ = λ∗u+


= −u+∂xu

Example: sensitivity to a scalar parameter

Generalizations

ν = U + 2icu

∇UA = −∂x

ut = (−ν∂x + γ∂xx + µ(x))� �� A

u

∇Uσ = Real(∇Uλ) ∇Uω = Imag(∇Uλ)

λ = σ + iω

complex Ginzburg-Landau

eigenvalue sensitivity

sensitivity of growth rate sensitivity of frequency

∇Uλ = u+∇UAu

Au = λu

A+u+ = λ∗u+



Example: choose base flow profile as control variable

p Reynolds number wave number base-flow

Reα, β

U(y)

∇Uλ = −(∇u)H u +∇u · u∗

relate mean flow modification to small control forces

delay onset of instabilities to higher Reynolds numbers; increase stability margins

Generalizations


Flow chart for sensitivity/receptivity analysis

Generalizations

base flow calculation

LNS adjoint LNS

adjoint modes global modes

receptivity sensitivity

Newton, Newton-Krylov selective frequency damping

power iteration Arnoldi

to baseflow modifications to localized feedback to structural changes

to external forces to mass sources/sinks to forcing a the wall


Generalizations


LNS adjoint LNS



Ubase Vbase

Example: flow around a cylinder


Generalizations

LNS adjoint LNS



(power iteration)


udirect vdirect



Generalizations

LNS adjoint LNS




vadjointuadjoint

(power iteration)



Generalizations

LNS adjoint LNS




uwavemaker vwavemaker



Generalizations

LNS adjoint LNS




uwavemaker vwavemaker


sensitivity to localized spatial feedback EPTT, Sao Paulo, Sept. 2012

Generalizations

LNS adjoint LNS



base flow calculation Example: flow around a cylinder

structural sensitivity

∇Uλ = −(∇u)H u +∇u · u∗


Generalizations

LNS adjoint LNS



base flow calculation Example: flow around a cylinder

structural sensitivity

∇Uλ = −(∇u)H u +∇u · u∗


place a small cylinder here to increase Rec to 70

Generalizations


What if the operator perturbation is stochastic ?

d

dtq − Lq = 0

L(t) = LS + �µ(t)S

dµ = −νµdt+ dW

statistically steady part uncertain part

stochastic process

ν ~ auto-correlation time

Generalizations


What if the operator perturbation is stochastic ?

We have to describe the solution statistically: propagation of covariance (second-order moments)

K = E(qqH)evolution equation for the covariance matrix (expansion of propagator A(t))

d

dtK = (LS + �2SD)K +K(LS + �2SD)H + �2(SKD +DKSH)

D =

� t

0exp(τLs)S exp(−τLS) exp(−ντ) dτwith

Generalizations


stochastic channel flow (perturbed base flow profile) ν = 1/5 � = 0.2

Re = 2000,α = 0.2,β = 2

nonlinear perturbation dynamics

Generalizations

adjoint analysis with check-pointing

the variational formulation also allows us to add nonlinear constraints to the cost functional

J = obj−�

q,

�d

dtq −N(q)

��→ max

nonlinear Navier-Stokes equations

How does this affect the adjoint looping ?



Generalizations


Example: nonlinear advective terms

first variation �u,u∇u� �−u∇u, δu�

We have direct terms appearing in the adjoint equation.


Adjoint equation is a variable-coefficient linear equation.


Generalizations


q(n)0 direct nonlinear problem

linear adjoint problem

u∇u

−u∇u

u(0) u(t)· · · · · ·· · ·the flow fields at the forward sweep have to be saved and injected into the backward sweep

checkpointing



Generalizations




u∇u

−u∇u

u(0) u(t)

For long-time integrations and high-dimensional problems we quickly reach the limits of storage devices.



Generalizations




u∇u

−u∇u

u(0) u(t)

store flow fields at coarse intervals and use as initial conditions for repeated forward integrations

optimized checkpointing

duplicate integration


no checkpointing (store everything)

Generalizations

total number of time-steps done

time

stored direct flow field



Generalizations


time




Generalizations


time



uniform checkpointing (store at a few equispaced locations)

Generalizations


time




Generalizations


time




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time

extra work




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time

extra work




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time

extra work

extra work

extra work




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time

extra work

extra work

extra work


six active checkpoints


Generalizations


time

extra work

extra work

extra work


five active checkpoints


Generalizations


time

extra work

extra work

extra work


four active checkpoints


Generalizations


time

extra work

extra work

extra work


three active checkpoints


Generalizations


time

extra work

extra work

extra work

better: binomial checkpointing best: minimal-repetition dynamic checkpointing


Generalizations


We often have governing equations with auxiliary evolution equations (e.g., for eddy-viscosity), but these auxiliary variables may not be part of the cost objective.

This leads to semi-norm constraints.

q =

�uνt

�d

dt

�uνt

�=

�f(u, νt)g(νt, ...)

�

�q� ≡ �u� �q� = 0 q �= 0

governing equations

turbulence model

not a true norm

We can have with

causes singularities (non-convergence)

Generalizations


We need additional constraints to avoid singularities.

constrained variational approach (penality terms)

optimization on hyper-spheres

constraint hyper-sphere

projection

Generalizations


Is the 2-norm appropriate for all applications ? Can we consider a worst-case scenario ?

dependence on chosen norm

�a�p = (|ax|p + |ay|p)1/p

p = 1 p = 2 p = 4 p = ∞

introduce a p-norm

Generalizations


localization of the optimal structures, symmetry breaking (work in progress)

p = 2 p = ∞

multiple inhomogeneous directions/complex geometry

Generalizations

global mode analysis

for most industrial applications we cannot assume the existence of homogeneous directions that can be treated by a Fourier transform

rather, the eigenfunction will depend on more than one inhomogeneous coordinate direction



Generalizations


q =

q1

q2...

qN

L ∈ CN×N L ∈ CN2×N2

q =

q1,1

q1,2...

qN,N

∼ N3 ∼ N6

state vector

stability matrix

operation count one inhomogeneous direction

two inhomogeneous directions



Generalizations


direct eigenvalue algorithms quickly become prohibitively expensive

iterative eigenvalue algorithms (Arnoldi technique) have to be used



Generalizations


≈L V

H

Arnoldi algorithm

action of the linear operator L is expressed within an orthonormal basis V

orthogonal basis

Hessenberg matrix

stability matrix

V

orthogonal basis

action of L action of H EPTT, Sao Paulo, Sept. 2012


Generalizations


≈L V

H

Arnoldi algorithm

represent the (large) stability matrix by a low-rank approximation based on an orthogonal basis

V H

orthogonal basis

Hessenberg matrix

stability matrix



Generalizations


qk = L qk−1

j = 1 : k − 1Hj,k−1 = �qj , qk�qk = qk −Hj,k−1 qj

Hk,k−1 = �qk�qk = qk/Hk,k−1

for

end

only multiplications by L are necessary

eig{L} ≈ eig{H}



Generalizations


≈L V

V H

stability matrix

computing global modes by diagonalizing H = DΛD−1

D D−1Λ

global modes



Generalizations


Examples of global modes: open cavity flow (two-dimensional)



Generalizations



global spectrum global mode



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Generalizations






Generalizations






Generalizations


Examples of global modes: jet in cross flow (three-dimensional)

global mode snapshot

jet

cross flow

Arnoldi algorithm


Arnoldi algorithm (a Krylov subspace technique) to compute the Hessenberg matrix H

Jacobian-free framework

DNS

ARPACK


Summary

  A variational approach for fluid stability problems provides a flexible and effective framework that allows the treatment time-invariant, time-periodic, time-dependent, linear and nonlinear problems.

  Adjoint variables can be interpreted as carriers of gradient/sensitivity information to external driving terms.

  Weighted (scalar) products of direct and adjoint variables yield structural sensitivity information and can be used for complex flow optimizations or the influence analysis of particular terms in the governing equations.

  Semi-norm constraints and p-norm extensions add even more flexibility.


Documents

A variaonal frameworkforﬂow op’mizaon :anintroduc’ontoadjointssites.poli.usp.br/org/eptt2012/Presentations/Schmid/EPTT_Lecture2.pdf · time-periodic and generally time-dependent